
theorem Th23: ::  WWA1:
  for R being DB-Rel holds Dependency-str R is full_family
proof
  let D be DB-Rel;
  set S = Dependency-str D;
  set T = the Attributes of D, R = the Relationship of D;
A1: now
    let A, B, C being Subset of T;
    assume that
A2: [A, B] in S and
A3: [B, C] in S;
A4: B >|> C, D by A3,Th10;
A5: A >|> B, D by A2,Th10;
    A >|> C, D
    proof
      let f, g being Element of R;
      assume f|A = g|A;
      then f|B = g|B by A5;
      hence thesis by A4;
    end;
    hence [A, C] in S;
  end;
  then
A6: S is (F2) by Th18;
A7: S is (DC3)
  proof
    let A, B being Subset of T such that
A8: B c= A;
    A >|> B, D
    proof
      let f, g being Element of R such that
A9:   f|A = g|A;
      thus f|B = (f|A)|B by A8,RELAT_1:74
        .= g|B by A8,A9,RELAT_1:74;
    end;
    hence thesis;
  end;
  hence S is (F1);
  thus S is (F2) by A1,Th18;
  thus S is (F3) by A7,A6;
  thus S is (F4)
  proof
    let A, B, A9, B9 being Subset of T;
    assume that
A10: [A, B] in S and
A11: [A9, B9] in S;
A12: A9 >|> B9, D by A11,Th10;
A13: A >|> B, D by A10,Th10;
    (A\/A9) >|> (B\/B9), D
    proof
      let f, g be Element of R;
      assume
A14:  f|(A\/A9) = g|(A\/A9);
      f|A9=(f|(A\/A9))|A9 by RELAT_1:74,XBOOLE_1:7
        .=g|A9 by A14,RELAT_1:74,XBOOLE_1:7;
      then
A15:  f|B9 = g|B9 by A12;
      f|A=(f|(A\/A9))|A by RELAT_1:74,XBOOLE_1:7
        .=g|A by A14,RELAT_1:74,XBOOLE_1:7;
      then
A16:  f|B = g|B by A13;
      thus f|(B\/B9)=f|B\/f|B9 by RELAT_1:78
        .= g|(B\/B9) by A16,A15,RELAT_1:78;
    end;
    hence thesis;
  end;
end;
